Finite Group Databasebeta

← #366306

Subgroups — #366306

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#1930C212yesmaximal, complemented
3#20844C211yesmaximal, characteristic
4#20599C211yesmaximal, characteristic
5C2×#1784C211yesmaximal, characteristic
6#20385C211yesmaximal, characteristic
7#11164C212yesmaximal, complemented
8#11145C212yesmaximal, complemented
9#10942C212yesmaximal, complemented
10#10727C211yesmaximal, characteristic
11#10723C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428C2212yes—
16C2×#419C2212yes—
17C2×#419C2212yes—
18C22×#103—28no—
19C22×#103—22no—
20C22×#103—22no—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#101C2212yes—
24C22×#101C2212yes—
25C2×#407C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30#2280C2211yescharacteristic
31#1930—28no—
32#1853—22no—
33#1852—22no—
34#1787—22no—
35#1787—22no—
36#1784C418yescomplemented
37#1752—22no—
38C8×C22⋊C4—22no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#341C2212yes—
42C2×#341C2212yes—
43#1736C2212yes—
44#1729C2211yescharacteristic
45#1729C2211yescharacteristic
46C23×Q8—21no—
47C23×Q8—22no—
48C23×Q8—24no—
49C23×Q8—24no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—42no—
55C23×Q8—42no—
56C23×Q8C2×C414yes—
57C23×Q8C2312yes—
58C24×C4—21no—
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C22×M16C2312yes—
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2312yes—
65C23×C8—22no—
66C23×C8—22no—
67C23×C8—24no—
68C23×C8—42no—
69C23×C8D412yes—
70C23×C8D412yes—
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4—22no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2312yes—
81#428—24no—
82#428—42no—
83C2×#107—22no—
84C2×#106—22no—
85#419—24no—
86#419—24no—
87#419—42no—
88#419—42no—
89C2×#103—216no—
90C2×#103—216no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103D418yes—
101C2×#101—24no—
102C2×#101—24no—
103C2×#101—42no—
104C2×#101—42no—
105C2×#101C2×C418yescomplemented
106C2×#101C2×C418yescomplemented
107#407—22no—
108#407—22no—
109#407—22no—
110#407—22no—
111#407C2×C414yes—
112#407C2×C414yes—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116#405—24no—
117#405—42no—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122#381—21no—
123#380D412yes—
124#380D412yes—
125#379—21no—
126C4×C22⋊C4—22no—
127#343—24no—
128#343—42no—
129#341—24no—
130#341—42no—
131#340—24no—
132#336—22no—
133#336—22no—
134C25#4111yescharacteristic
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—28no—
140C22×Q8—216no—
141C22×Q8—28no—
142C22×Q8—216no—
143C22×Q8—28no—
144C22×Q8—28no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—44no—
154C22×Q8—44no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—48no—
159C22×Q8—48no—
160C22×Q8—48no—
161C22×Q8—48no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—48no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×Q8C22×C414yes—
170C22×Q8C2×D414yes—
171C22×Q8#4114yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—84no—
186C23×C4C22×C412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4#4111yescharacteristic
190C23×C4#4112yes—
191C23×C4#4112yes—
192C23×C4#4112yes—
193C23×C4C2411yescharacteristic
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16C22×C412yes—
212C2×M16C22×C414yes—
213C2×M16C22×C412yes—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—22no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—22no—
226C22×C8—24no—
227C22×C8—24no—
228C22×C8—22no—
229C22×C8—42no—
230C22×C8—42no—
231C22×C8—48no—
232C22×C8—48no—
233C22×C8C2×D412yes—
234C22×C8C2×D414yes—
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237#117—22no—
238#117—22no—
239#117—22no—
240#117—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C42—21no—
260C2×C42—21no—
261#107—44no—
262#106—44no—
263#103—216no—
264#103—28no—
265#103—28no—
266#103—44no—
267#103—44no—
268#103—48no—
269#101—216no—
270#101—48no—
271C22⋊C8—28no—
272C22⋊C8—28no—
273C22⋊C8—44no—
274#97—42no—
275#97—42no—
276C4×C8—42no—
277C4×C8—42no—
278#95—22no—
279#95—22no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—24no—
284C24#11712yes—
285C24#12112yes—
286C24#12612yes—
287C24C2×#4111yescharacteristic
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—232no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—416no—
305C2×Q8—432no—
306C2×Q8—48no—
307C2×Q8—416no—
308C2×Q8—48no—
309C2×Q8—416no—
310C2×Q8—416no—
311C2×Q8—432no—
312C2×Q8—416no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—48no—
326C2×Q8C4×D418yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—416no—
351C22×C4—48no—
352C22×C4—416no—
353C22×C4—48no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—816no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4#11712yes—
367C22×C4C4×D414yes—
368C22×C4C4×D414yes—
369C22×C4#12112yes—
370C22×C4#12314yes—
371C22×C4#12414yes—
372C22×C4#12612yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)12yes—
383M16—24no—
384M16—24no—
385M16—24no—
386M16—24no—
387M16—42no—
388M16—42no—
389M16—42no—
390M16—42no—
391M16—42no—
392M16—42no—
393M16—42no—
394M16—42no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—48no—
408C2×C8—48no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8#12012yes—
421C2×C8#12012yes—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—48no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C23—28no—
439C23—28no—
440C23—216no—
441C23—24no—
442C23—22no—
443C23—28no—
444C23—28no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23#38814yes—
452C23#47114yes—
453C23#47314yes—
454C23C2×#11711yescharacteristic
455C23C2×#12111yescharacteristic
456C23C2×#12611yescharacteristic
457C23#54612yes—
458C23#55112yes—
459Q8—28no—
460Q8—28no—
461Q8—432no—
462Q8—44no—
463Q8—44no—
464Q8—44no—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—432no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4—816no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4#38814yes—
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×C4×D412yes—
502C2×C4#51812yes—
503C2×C4C2×#12011yescharacteristic
504C2×C4C2×#12311yescharacteristic
505C2×C4#52512yes—
506C2×C4#52511yescharacteristic
507C2×C4C2×#12411yescharacteristic
508C2×C4#53411yescharacteristic
509C2×C4#53511yescharacteristic
510C2×C4#53812yes—
511C2×C4D4212yes—
512C2×C4#54612yes—
513C2×C4#54612yes—
514C2×C4#54612yes—
515C2×C4#55112yes—
516C2×C4#55112yes—
517C8—22no—
518C8—22no—
519C8—42no—
520C8—44no—
521C8—42no—
522C8—42no—
523C8—82no—
524C22—216no—
525C22—216no—
526C22—22no—
527C22—22no—
528C22—24no—
529C22—24no—
530C22—22no—
531C22—24no—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22#193018yescomplemented
538C22C2×#38811yescharacteristic
539C22#234212yes—
540C22#240811yescharacteristic
541C22#248012yes—
542C22#253411yescharacteristic
543C22C2×#47112yes—
544C22C2×#47312yes—
545C4—24no—
546C4—48no—
547C4—81no—
548C4—84no—
549C4—81no—
550C4—82no—
551C4C4×#12011yescharacteristic
552C4#232111yescharacteristic
553C4#235812yes—
554C4#240511yescharacteristic
555C4#247212yes—
556C4#248211yescharacteristic
557C2—22no—
558C2—28no—
559C2—22no—
560C2#1399111yescharacteristic
561C2C2×#193012yes—
562C2#2427811yescharacteristic
563C2#2428211yescharacteristic
564C2#2813512yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1930C2semidirect
7. #11164C2semidirect
8. #11145C2semidirect
9. #10942C2semidirect
36. #1784C4semidirect
105. C2×#101C2×C4semidirect
106. C2×#101C2×C4semidirect
537. C22#1930semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×#1032
C2×#19302
#208442
#205992
C2×#17842
#203852
#111642
#111452
#109422
#107272
#107232